Abstract
The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:
where λ > 0 is a real parameter, Ω is a bounded domain in RN, with boundary ∂Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (−Δ) p su is the fractional p-Laplacian operator of u and, similarly, (−Δ) q sv is the fractional q-Laplacian operator of v. Since possibly p ≠q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ1 for a related system, they prove that there exists a positive solution for the problem when λ < λ1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ1-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ1.
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Franzina, G. and Palatucci, G., Fractional p-eigenvalues, Riv. Math. Univ. Parma, 5(2), 2014, 373–386.
Iannizzotto, A. and Squassina, M., Weyl-type laws for fractional p-eigenvalue problems, Asymptot. Anal., 88(4), 2014, 233–245.
Brown, K. J. and Zhang, Y., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193(2), 2003, 481–499.
Chen, W. and Deng, S., The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Anal. Real World Appl., 27, 2016, 80–92.
Zhang, G., Liu, X. and Liu, S., Remarks on a class of quasilinear elliptic systems involving the (p, q)-Laplacian, Electron. J. Differential Equations, 2005(20), 2005, 10 pages.
Goyal, S. and Sreenadh, K., Existence of multiple solutions of p-fractional Laplace operator with signchanging weight function, Adv. Nonlinear Anal., 4(1), 2015, 37–58.
Fiscella, A., Pucci, P. and Saldi, S., Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators, Nonlinear Anal., 158(2), 2017, 109–131.
Di Castro, A., Kuusi, T. and Palatucci, G., Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(5), 2016, 1279–1299.
Di Nezza, E., Palatucci, G. and Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136(5), 2012, 521–573.
Grisvard, P., Elliptic Problems in Nonsmooth Domains, 2nd ed., With a Foreword by Susanne C. Brenner, Classics in Applied Mathematics, 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
Pucci, P., Xiang, M. and Zhang, B., Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN, Calc. Var. Partial Differential Equations, 54(3), 2015, 2785–2806.
Drabek, P., Stavrakakis, N. M. and Zographopoulos, N. B., Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations, 16(12), 2003, 1519–1531.
Amghibech, S., On the discrete version of Picone’s identity, Discrete Appl. Math., 156(1), 2008, 1–10.
Mosconi, S. and Squassina, M., Nonlocal problems at nearly critical growth, Nonlinear Anal., 136, 2016, 84–101.
Del Pezzo, L. M. and Quaas, A., A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations, 263(1), 2017, 765–778.
Acknowledgement
This paper was started while Y. Fu was visiting the Dipartimento di Matematica e Informatica of the Universit`a degli Studi di Perugia, Italy, in June and July 2016 and was completed when P. Pucci was visiting the Department of Mathematics of the Harbin Institute of Technology at Harbin, China, in July 2017. Both authors thank the departments for the hospitality.
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Dedicated to Professor Philippe G. Ciarlet on the occasion of his 80th birthday, with high feelings of esteem for his notable contributions in mathematics and great affection
This work was supported by the National Natural Science Foundation of China (No. 11771107), the Italian MIUR Project Variational Methods, with Applications to Problems in Mathematical Physics and Geometry (No. 2015KB9WPT 009), the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and the INdAMGNAMPA Project 2017 titled Equazioni Differenziali non lineari (No. Prot 2017 0000265).
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Fu, Y., Li, H. & Pucci, P. Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p, q)-Laplacian Operators. Chin. Ann. Math. Ser. B 39, 357–372 (2018). https://doi.org/10.1007/s11401-018-1069-1
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DOI: https://doi.org/10.1007/s11401-018-1069-1